Foundations of MSE - Exam 1 Definitions

On Lectures 1-8 (really 2-8)

Solid

A solid is a sample of matter that can retain its volume and shape,

Liquid

Can retain its volume but not rigid

Gas

Cannot retain its volume and is not rigid

Atoms

The elementary chemical constituents of matter.

They cannot be broken during a chemical reaction.

They can interact with one another more or less strongly to form materials

Atoms are made of

Protons and neutrons, surrounded by electrons

What does X, A and Z represent?

X- atomic symbol

A- mass number (number of protons and neutrons)

Z - atomic number (number of protons only)

Identify how many protons, neutrons and electrons are in this system

P⁺ 29

N⁰ 34

e^- 29

Mass Defect

Mass of atom isn’t exactly equal to the masses of its isolated subatomic constituents

Atomic units (definition)

Used to simplify calculations such as the Schrodinger Equation

Definition of angular momentum

L = mrv_θ

When is the angular momentum a constant of motion?

When force acting on particle is radial only with no angular component (F_θ=0)

Total energy as a function of KE + PE in polar coordinates

½ * m_e (r_nθ)² - e²/ 4πε₀ r_n = R_∞/n²

Velocity in polar coordinates

v = dr/dt r̂ + r dθ/dt θ̂

What does Bohr’s Model not explain?

• Why can energies and radii only have specific values E_n and r_n?

• What is the reason for orbit shapes?

How are electrons represented in Bohr’s Model vs. Schrodinger’s Model?

Bohr: Electrons are points

Schrodinger: Electrons are clouds

In which model are position and momentum precisely known?

Bohr’s Model

In which model are position and momentum averages known?

Schrodinger’s Model

What defines the electron state and electron energy E in Schrodinger’s Model?

Wave function Ψ

What defines the electron state and energy E for the Bohr Model?

p, r

Wave function Ψ( r ) definition

Wave function of an electron is the function of the space variable r describing the state of the electron

The values of the wave function are

Complex

Square modulus of wave function

Gives probability density

Square Modulus of Ψ (r )

ρ ( r) = |Ψ (r )|²

Normalization condition

Total probability (of where electron must be in space) = 1

How do we find the specific values of the total energy E = K + U (for quantum mechanics)

Schrodinger’s Equation

General Schrodinger’s Equation (full)

kinetic portion of SE

potential part of SE

Schrodinger’s Equation for ₁¹H

Schrodinger Equation for ₁¹H in Spherical coordinates

Spherical harmonics

Special functions defined on the surface of a sphere, used in different fields

Spherical harmonic of order l is an angular function Y(θ, Φ) such that…

∇² (r^lY(θ,Φ)) = 0

For each value of l = 0, 1, etc. there exist _____ independent solutions of ________ that are labeled by index ___ ranging from _____ to ______

2l + 1

Y_lm(θ,Φ)

m

-l to l

Nodal surface

Where function is 0

A positive sign for m means an _____ orientation for Ylm

x-axis

Kinetic energy portion of ¹₁H S.E.

Potential Energy portion for ¹₁H S.E.

Hybridized molecular orbital (ψ (r )) written as…

Way to write out the minimization of energy in a generalized eigenvalue form

What is H?

Hamiltonian Matrix of Orbital Interaction Coefficients

What is S?

The matrix of orbital overlap coefficients

Born-Oppenheimer Approximation for H₂⁺

Which part of this equation is the quantum energy and which part is classical?

Quantum: First

Classical: 2nd

Generalized Born-Oppenheimer Approximation for System with 2+ Nuclei

What does ||R_i - R_j|| represent in the B-O Equation?

Distance between nuclei i and j

What does Z_i and Z_j in the B-O Approximation Represent

Nuclei charge

What kind of orbital is this, and will this molecule be stable?

Bonding Orbital, yes

What kind of orbital is this, and will this molecule be stable?

Anti-bonding

No

Equation to determine orbit radii r_n (Bohr)

rn = a_on²

A material is made of these 2 subatomic particles

nuclei and electrons

Hybridized Orbital

Mixture of orbitals of atoms that form a covalent bond

Hybridized molecular orbital equation

Ψ(n) = c₁Φ₁ + c₂Φ₂

How can the minimization of energy be written?

H (c1 c2) = ES (c1 c2)

H is what

Hamiltonian

E is what

Overlap integral matrix

Hybridization of H₂⁺

1s orbitals form 1 bonding and 1 antibonding orbital

sp, sp2, sp3 orbitals good for?

Linear, hexagonal, or tetrahedral bonding

Ionic bonding

2 molecules of differing electronegativity (less EN gives electron to more EN)

electronegativity

ability to attract/retain electrons

When does electronegativity increase?

As energies of outer-most orbitals are more negative

What can happen to orbitals when there are electronegativity differences (ex. HF)

Orbitals will favor characteristics 1 molecule over another

Covalent bonding

Electrons shared with directional preference

Ionic bonding

electrons are localized around atoms but unequally distributed

Metallic bonding

Electrons are shared and delocalized without directional preference

VDW bonding

Electrons are localized around atoms and are equally distributed

Strongest to weakest bonds

1. covalent

2. ionic

3. metallic

4. VDW

Spherical harmonics Y(θ, Φ)

Angular functions that model lots of processes

Angular function Y(θ, Φ) such that

∇² (r^l (Y(θ, Φ)) = 0

Radical Orbital Function equation

What does a radial orbital function calculate?

Exact probability of locating an electron at a specific distance from the atom’s nucleus

Number of nodes is where R_nl(r ) goes to 0 is represented by what equation?

n-l-1

As r —> 0, what happens to R_nl(r )?

is proportional to r^l

Hydrogen-like atoms

1-electron atom with nuclear charge Z>1, net charge of Z-1 and are denoted by ^A_ZX^(Z-1)+

Nuclear shielding

Electron-electron repulsion leads to nucleus attraction to be WEAKER

How is nuclear shielding calculated?

= Z^effε₀ / n²

Electron spin

Magnetic moment carried by electron, can take on 2 values +- µ_b

Electron pattern for quantized system

Electron pattern for non-quantized system

Building-up principle

To get electron configuration

1. lowest E to highest E

2. no more than 2 e- in orbital

3. Separate on orbital level first

Order of orbital levels

1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p

What do orbital levels help with

To estimate electronic configuration of atom by approximating the nuclear shielding effect

What is a symmetry operation?

Mathematical operation that maps an object back onto itself

What kind of symmetry is this?

Mirror

What kind of symmetry is this

Rotational

What kind of symmetry is this?

Translational

Which symmetry operation defines a lattice?

Translation

Translation vector t connects any 2 lattice points through this equation

t = ua + vb + wc

where u, v, w are the scalar translation and a, b, c are lattice vectors

Lattice (τ)

Set of translation vectors for all possible values of u, v, w and thus infinite

τ =

{ t | t = ua + vb + wc (u, v, w integers) }

Unit Cell

Smallest repeat building block of a crystal that can be replicated across an entire crystal

Lattice parameters of unit cell

[a, b, c, alpha, beta, gamma]

4 unit cell types

1. primitive

2. body-centered

3. face-centered

4. side-centered

Crystallographic Restriction Theorem

If a rotational symmetry of an angle α is permitted in a lattice, then:

cos α = M/2

Where m is an integer

What does the CRT restrict

geometries to 2, 3, 4, or 6-fold rotation

M values of CRT

-2, -1, 0, 1, 2

Why are quasicrystals not considered crystals? I.e. why do they fail the CRT

5-fold means that M is not an integer

Quasicrystals violate what

Translational symmetry

Volume of primitive unit cell

V/N = V*

V = volume, N = # of lattice points

Spatial Lattice vs Crystal Structure

Lattice - mathematical array of points, where every point has identical surroundings

Crystal Structure - physical material created by placing a motif (such as atom) at every lattice point